March 22, 2008

Far far far away from the last post on compressive sensing, two other talks closer to every day problems with standard data (orthogonal matrices and quantization), which i have been watching during breakfast (quite a long one).

Ofer Zeitouni: A Correlation Inequality for Nonlinear Reconstruction, on the question whether the Karhunen-Loève transform remains optimal in the non-linear approximation for gaussian vectors. Karhunen-Loève transform (or Proper orthogonal decomposition, or Principal component analysis) is a central tool in statistics (often named after Hotelling) and signal processing, yielding "optimal" orthogonal transforms for uncorrelated data. The duet formed by Kari Karhunen and Michel Loève (even a triplet which Harold Hotelling) reassembles to an Ever Alone Hunk.

Zhidong Bai: Statistical Analysis for Rounding Data. Most of the discrete versions of the continuous setting deal with the discretization of time or space, i.e. sampling. Now what happens to the discrete amplitudes, i.e. discretization in the "value" domain? Especially that standard estimation of mean and variance are not consistant. Some counter-intuitive experiments with respect to the central limit theorem. Here, the larger, the worse.

This first short note on Compressive Sensing is, of course, anagram-driven since it has been applied for some Vein Mess Processing (see the images on the last page of Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging, by Michael Lustig, David Donoho and John M. Pauly). Some matlab code for compressed sensing MRI is made available. It is also dedicated to Igor Carron who carefully blogs and extracts information bits from the overwhelming litterature on CS (see Compressive Sensing Resources). For short, compressed or compressive sensing aims at saving bits by sampling and compressing structured signals at the same time, allowing some potential error. It is strongly related to sparse decompositions. An open challenge: find the sparsest/most compressed, but also most elegant, definition for "compressive sensing"! I do consider the four-word Wedderburn theorem "every finite field commutates" (in French: "tout corps fini commute", with open poetic interpretations) one of the beautiful mantras (sure, allowing an amount of metonymia, five-word versions are more correct). The right-side image of a "corps fini" is borrowed from here.

Compressive sensing may solve some problems. Hope it could also contribute to stock exchange troubles by improving SEC Senses. While standard banks fail, filter banks (and wavelets of course) never deceive.

March 6, 2008

Fortunately, the wavelet activity is not as sparse as the representation it provides. Indeed, there is no such thing as the lazy wavelet (l'ondelette indolente). Recent additions from the wavelet world, namely wavelet names in "*let" include treelets, noiselets, needlets and a connection to Vassili Strela multiwavelet toolbox, which was long unavailable. Wavelets do not suffer (inactivity). The anagramatic connection linking "ondelette" "et dolente ?" is thus... wrong. There is such thing as the lazy wavelet. Contradiction? No, we encourage star-let submission!